{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T18:59:16Z","timestamp":1790621956840,"version":"4.1.0"},"reference-count":43,"publisher":"Public Library of Science (PLoS)","issue":"9","license":[{"start":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T00:00:00Z","timestamp":1790553600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union under the REFRESH \u2013 Research Excellence For Region Sustainability and High-tech Industries","award":["CZ .10.03.01\/00\/22_003\/0000048"],"award-info":[{"award-number":["CZ .10.03.01\/00\/22_003\/0000048"]}],"id":[{"id":"https:\/\/ror.org\/019w4f821","id-type":"ROR","asserted-by":"crossref"}]}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"\n We develop a unified fixed point theory for enriched bivariate contractions in ordered Banach spaces and in ordered convex metric spaces. Two classes of mappings are introduced and analysed: enriched bivariate Ciric-Reich-Rus contractions (\n \n \n \n \n \n \n \ud835\udc9e<\/mml:mi>\n \n B<\/mml:mi>\n C<\/mml:mi>\n R<\/mml:mi>\n R<\/mml:mi>\n C<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) and enriched bivariate interpolative Ciric-Reich-Rus contractions (\n \n \n \n \n \n \n \ud835\udc9e<\/mml:mi>\n \n B<\/mml:mi>\n I<\/mml:mi>\n C<\/mml:mi>\n R<\/mml:mi>\n R<\/mml:mi>\n C<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ). For each class we prove the existence and uniqueness of coupled fixed points and establish the geometric convergence of the coupled Krasnoselskij iteration. The methodological contribution is a transparent product space reduction: the coupled averaged operator associated with an enriched bivariate map is shown to be an ordinary (single variable) Ciric-Reich-Rus operator on the product space, so that the classical theory applies directly and every algebraic step is explicit. This reduction yields a sharp sufficient condition for the existence of a coupled fixed point, namely\n \n \n \n \n \n (<\/mml:mo>\n 2<\/mml:mn>\n a<\/mml:mi>\n +<\/mml:mo>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n \/<\/mml:mo>\n (<\/mml:mo>\n \n k<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n +<\/mml:mo>\n 2<\/mml:mn>\n b<\/mml:mi>\n <<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n a<\/jats:italic>\n ,\n b<\/jats:italic>\n are the contraction constants and\n \n \n \n \n \n \n k<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are the enrichment constants. In particular the coupling doubles the coefficient of the point distance term relative to the univariate enriched theory, so a strictly stronger requirement on\n a<\/jats:italic>\n is unavoidable. We further show that the condition 2\n a<\/jats:italic>\n \u2009+\u20093\n b<\/jats:italic>\n \u2009<\u20091 stated in earlier formulations is sufficient but not necessary, and we exhibit an explicit map for which a coupled fixed point exists while 2\n a<\/jats:italic>\n \u2009+\u20093\n b<\/jats:italic>\n \u2009<\u20091 fails. The abstract results are applied first to a Volterra type coupled integral system, and then to the fractional order Thomas cyclically symmetric attractor formulated with the Caputo-Fabrizio derivative, where an averaged mapping yields local in time existence and uniqueness even though the long time dynamics are chaotic. All numerical results are produced by direct computation. A bifurcation diagram and a largest Lyapunov exponent spectrum, computed independently, agree window for window and confirm a period doubling route to chaos with intermediate periodic windows, while the measured convergence factor of the coupled iteration is geometric.\n <\/jats:p>","DOI":"10.1371\/journal.pone.0358480","type":"journal-article","created":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T17:55:21Z","timestamp":1790618121000},"page":"e0358480","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["A unified fixed point approach for enriched bivariate mappings with applications to volterra systems and fractional-order chaos"],"prefix":"10.1371","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5423-4596","authenticated-orcid":true,"given":"Khaleel","family":"Ahmad","sequence":"first","affiliation":[{"name":"University of Management and Technology","place":["Lahore, Pakistan"],"department":["Department of Mathematics"]},{"name":"Khazar University","place":["Baku, Azerbaijan"],"department":["Center for Theoretical Physics"]},{"name":"The University of Lahore","place":["Lahore, Pakistan"],"department":["Department of Mathematics and Statistics"]},{"name":"Biruni University","place":["Istanbul, Turkey"],"department":["Software Development Department"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sahib","family":"Yar","sequence":"additional","affiliation":[{"name":"University of Southern Punjab","place":["Multan, Punjab, Pakistan"],"department":["Department of Mathematics and Statistics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdul Rahim","family":"Khan","sequence":"additional","affiliation":[{"name":"University of Southern Punjab","place":["Multan, Punjab, Pakistan"],"department":["Department of Mathematics and Statistics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nida","family":"Najeeb","sequence":"additional","affiliation":[{"name":"University of Southern Punjab","place":["Multan, Punjab, Pakistan"],"department":["Department of Mathematics and Statistics"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adil","family":"Jhangeer","sequence":"additional","affiliation":[{"name":"VSB-Technical University of Ostrava","place":["Ostrava-Poruba, Czech Republic"],"department":["IT4Innovations"]},{"name":"Biruni University","place":["Istanbul, Turkey"],"department":["Department of Computer Engineering"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Walid","family":"Abdelfattah","sequence":"additional","affiliation":[{"name":"Northern Border University","place":["Arar, Saudi Arabia"],"department":["Humanities and Social Research Center"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,9,28]]},"reference":[{"issue":"1","key":"pone.0358480.ref001","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leur application aux equations integrales","volume":"3","author":"S Banach","year":"1922","journal-title":"Fundam Math"},{"key":"pone.0358480.ref002","first-page":"71","article-title":"Some results on fixed points","volume":"60","author":"R Kannan","year":"1968","journal-title":"Bull Calcutta Math Soc"},{"issue":"4","key":"pone.0358480.ref003","first-page":"405","article-title":"Some results on fixed points II","volume":"76","author":"R Kannan","year":"1969","journal-title":"Am Math Mon"},{"issue":"2","key":"pone.0358480.ref004","doi-asserted-by":"crossref","first-page":"85","DOI":"10.31197\/atnaa.431135","article-title":"Revisiting the Kannan type contractions via interpolation","volume":"2","author":"E Karapi\u0307nar","year":"2018","journal-title":"Advances in the Theory of Nonlinear Analysis and its Application"},{"issue":"11","key":"pone.0358480.ref005","doi-asserted-by":"crossref","first-page":"256","DOI":"10.3390\/math6110256","article-title":"Interpolative Reich-Rus-Ciric type contractions on partial metric spaces","volume":"6","author":"E Karapinar","year":"2018","journal-title":"Mathematics"},{"issue":"2","key":"pone.0358480.ref006","doi-asserted-by":"crossref","DOI":"10.1007\/s11784-020-0769-9","article-title":"Approximating fixed points of enriched contractions in Banach spaces","volume":"22","author":"V Berinde","year":"2020","journal-title":"J Fixed Point Theory Appl"},{"issue":"2","key":"pone.0358480.ref007","doi-asserted-by":"crossref","first-page":"173","DOI":"10.37193\/CJM.2021.02.03","article-title":"Fixed point theorems for enriched \u0106iri\u0107-Reich-Rus contractions in Banach spaces and convex metric spaces","volume":"37","author":"V Berinde","year":"2021","journal-title":"CJM"},{"key":"pone.0358480.ref008","doi-asserted-by":"crossref","first-page":"113217","DOI":"10.1016\/j.cam.2020.113217","article-title":"Fixed point theorems for Kannan type mappings with applications to split feasibility and variational inequality problems","volume":"386","author":"V Berinde","year":"2021","journal-title":"J Comput Appl Math"},{"issue":"4","key":"pone.0358480.ref009","doi-asserted-by":"crossref","DOI":"10.1007\/s11784-021-00904-x","article-title":"Approximating fixed points of enriched Chatterjea contractions by Krasnoselskij iterative algorithm in Banach spaces","volume":"23","author":"V Berinde","year":"2021","journal-title":"J Fixed Point Theory Appl"},{"issue":"3","key":"pone.0358480.ref010","doi-asserted-by":"crossref","first-page":"498","DOI":"10.3390\/sym13030498","article-title":"Existence and Approximation of Fixed Points of Enriched Contractions and Enriched \u03c6-Contractions","volume":"13","author":"V Berinde","year":"2021","journal-title":"Symmetry"},{"issue":"2","key":"pone.0358480.ref011","doi-asserted-by":"crossref","first-page":"459","DOI":"10.3934\/naco.2023022","article-title":"Fixed point theorems of enriched Ciric\u2019s type and enriched Hardy-Rogers contractions","volume":"15","author":"Anjali","year":"2025","journal-title":"NACO"},{"issue":"2","key":"pone.0358480.ref012","doi-asserted-by":"crossref","first-page":"391","DOI":"10.4995\/agt.2022.16701","article-title":"Fixed point results of enriched interpolative Kannan type operators with applications","volume":"23","author":"M Abbas","year":"2022","journal-title":"Appl Gen Topol"},{"issue":"1","key":"pone.0358480.ref013","first-page":"1","article-title":"Fixed point results of R-enriched interpolative Kannan pair in R-convex metric spaces","volume":"32","author":"M Abbas","year":"2023","journal-title":"Creat Math Inform"},{"issue":"8","key":"pone.0358480.ref014","doi-asserted-by":"crossref","first-page":"2485","DOI":"10.2298\/FIL2108485A","article-title":"Common fixed point theorem for modified Kannan enriched contraction pair in Banach spaces and its applications","volume":"35","author":"R Anjum","year":"2021","journal-title":"Filomat"},{"issue":"16","key":"pone.0358480.ref015","doi-asserted-by":"crossref","first-page":"5455","DOI":"10.2298\/FIL2316455R","article-title":"Approximation and existence of fixed points via interpolative enriched contractions","volume":"37","author":"S Rawat","year":"2023","journal-title":"Filomat"},{"key":"pone.0358480.ref016","first-page":"18","article-title":"Fixed points via interpolative enriched contractions in metric and quasi-metric spaces","volume":"14","author":"M Edraoui","year":"2024","journal-title":"Adv Fixed Point Theory"},{"issue":"7","key":"pone.0358480.ref017","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1016\/j.na.2005.10.017","article-title":"Fixed point theorems in partially ordered metric spaces and applications","volume":"65","author":"TG Bhaskar","year":"2006","journal-title":"Nonlinear Analysis: Theory, Methods & Applications"},{"issue":"12","key":"pone.0358480.ref018","doi-asserted-by":"crossref","first-page":"4341","DOI":"10.1016\/j.na.2008.09.020","article-title":"Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces","volume":"70","author":"V Lakshmikantham","year":"2009","journal-title":"Nonlinear Analysis: Theory, Methods & Applications"},{"issue":"15","key":"pone.0358480.ref019","doi-asserted-by":"crossref","first-page":"4889","DOI":"10.1016\/j.na.2011.03.032","article-title":"Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces","volume":"74","author":"V Berinde","year":"2011","journal-title":"Nonlinear Analysis: Theory, Methods & Applications"},{"issue":"1","key":"pone.0358480.ref020","doi-asserted-by":"crossref","DOI":"10.1186\/1687-1812-2014-149","article-title":"Coupled solutions for a bivariate weakly nonexpansive operator by iterations","volume":"2014","author":"V Berinde","year":"2014","journal-title":"Fixed Point Theory Appl"},{"issue":"1","key":"pone.0358480.ref021","article-title":"Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations","volume":"2013","author":"N Hussain","year":"2013","journal-title":"Fixed Point Theory Appl"},{"issue":"1","key":"pone.0358480.ref022","doi-asserted-by":"crossref","DOI":"10.1186\/1687-1812-2013-50","article-title":"Discussion on some coupled fixed point theorems","volume":"2013","author":"B Samet","year":"2013","journal-title":"Fixed Point Theory Appl"},{"issue":"1","key":"pone.0358480.ref023","first-page":"29","article-title":"Remarks on some coupled coincidence point results in partially ordered metric spaces","volume":"20","author":"S Radenovic","year":"2014","journal-title":"Arab J Math Sci"},{"issue":"2","key":"pone.0358480.ref024","doi-asserted-by":"crossref","first-page":"142","DOI":"10.2996\/kmj\/1138846111","article-title":"A convexity in metric space and nonexpansive mappings","volume":"22","author":"W Takahashi","year":"1970","journal-title":"Kodai Math J"},{"issue":"01","key":"pone.0358480.ref025","doi-asserted-by":"crossref","first-page":"103","DOI":"10.56827\/JRSMMS.2022.1001.9","article-title":"A fixed point theorem on enriched (\u03c8, \u03c6\u03bb)-weakly contractive maps in convex metric spaces","volume":"10","author":"GVR Babu","year":"2022","journal-title":"JRSMMS"},{"key":"pone.0358480.ref026","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2022\/5695286","article-title":"Stability results for enriched contraction mappings in convex metric spaces","volume":"2022","author":"R Panicker","year":"2022","journal-title":"Abstr Appl Anal"},{"issue":"12","key":"pone.0358480.ref027","doi-asserted-by":"crossref","first-page":"815","DOI":"10.3390\/axioms13120815","article-title":"Approximating Fixed Points via Hybrid Enriched Contractions in Convex Metric Space with an Application","volume":"13","author":"B Rani","year":"2024","journal-title":"Axioms"},{"issue":"4","key":"pone.0358480.ref028","first-page":"2392","article-title":"Best proximity points for weak proximal enriched G contraction in graphical convex metric spaces","volume":"15","author":"C Li","year":"2025","journal-title":"J Appl Anal Comput"},{"key":"pone.0358480.ref029","doi-asserted-by":"crossref","first-page":"116064","DOI":"10.1016\/j.cam.2024.116064","article-title":"Common fixed point theorems for several pairs of generalized enriched contractions in a complete generalized convex metric space","volume":"451","author":"Y Ju","year":"2024","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"2","key":"pone.0358480.ref030","doi-asserted-by":"crossref","first-page":"5826","DOI":"10.29020\/nybg.ejpam.v18i2.5826","article-title":"Common fixed points of asymptotically regular mappings in convex metric spaces with application","volume":"18","author":"AR Khan","year":"2025","journal-title":"Eur J Pure Appl Math"},{"key":"pone.0358480.ref031","first-page":"1","article-title":"Enriched bivariate contraction map for approximating coupled operators with applications","volume":"23","author":"OT Wahab","year":"2023","journal-title":"Appl Math E-Notes"},{"issue":"1","key":"pone.0358480.ref032","first-page":"45","article-title":"On theta quasi Geraghty contractive mappings and application to perturbed Volterra and hypergeometric operators","volume":"63","author":"OT Wahab","year":"2023","journal-title":"Kyungpook Math J"},{"issue":"3","key":"pone.0358480.ref033","doi-asserted-by":"crossref","first-page":"3969","DOI":"10.1007\/s40808-024-01987-4","article-title":"Mathematical modeling and optimal control analysis of Monkeypox virus in contaminated environment","volume":"10","author":"W Okongo","year":"2024","journal-title":"Model Earth Syst Environ"},{"key":"pone.0358480.ref034","article-title":"Modeling and control of hepatitis B virus transmission dynamics using fractional order differential equations","volume":"2023","author":"NS Aguegboh","year":"2023","journal-title":"Commun Math Biol Neurosci"},{"issue":"1","key":"pone.0358480.ref035","doi-asserted-by":"crossref","first-page":"20240009","DOI":"10.1515\/cmb-2024-0009","article-title":"Existence and uniqueness of solution for a fractional hepatitis B model","volume":"13","author":"NS Aguegboh","year":"2025","journal-title":"Comput Math Biophys"},{"issue":"2","key":"pone.0358480.ref036","doi-asserted-by":"crossref","DOI":"10.1007\/s40808-024-02273-z","article-title":"A novel approach to modeling malaria with treatment and vaccination as control strategies in Africa using the Atangana\u2013Baleanu derivative","volume":"11","author":"NS Aguegboh","year":"2025","journal-title":"Model Earth Syst Environ"},{"key":"pone.0358480.ref037","doi-asserted-by":"crossref","first-page":"100523","DOI":"10.1016\/j.rico.2025.100523","article-title":"Fractional optimal control problem modeling bovine tuberculosis and rabies co-infection","volume":"18","author":"B Diallo","year":"2025","journal-title":"Results in Control and Optimization"},{"key":"pone.0358480.ref038","article-title":"Modeling the dynamics of onchocerciasis using the Atangana-Baleanu fractional order with control measures","volume":"28","author":"NS Aguegboh","year":"2025","journal-title":"Sci Afr"},{"issue":"8","key":"pone.0358480.ref039","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14445\/22315373\/IJMTT-V67I8P501","article-title":"A fractional order mathematical model for the dynamics of chicken pox","volume":"67","author":"NS Aguegboh","year":"2021","journal-title":"Int J Math Trends Technol"},{"key":"pone.0358480.ref040","article-title":"Fractional order modelling study","author":"Scientific African","year":"2025","journal-title":"Sci Afr"},{"issue":"2","key":"pone.0358480.ref041","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1090\/S0002-9939-1977-0436113-5","article-title":"Fixed point theorems for mappings with a contractive iterate at a point","volume":"62","author":"J Matkowski","year":"1977","journal-title":"Proc Am Math Soc"},{"issue":"10","key":"pone.0358480.ref042","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1142\/S0218127499001383","article-title":"Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, \u201clabyrinth chaos\u201d","volume":"09","author":"R Thomas","year":"1999","journal-title":"Int J Bifurcation Chaos"},{"issue":"2","key":"pone.0358480.ref043","first-page":"1","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"M Caputo","year":"2015","journal-title":"Prog Fract Differ Appl"}],"container-title":["PLOS One"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0358480","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T17:55:29Z","timestamp":1790618129000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0358480"}},"subtitle":[],"editor":[{"given":"Vikash Kumar","family":"Sinha","sequence":"first","affiliation":[{"name":"Srinath University","place":["INDIA"]}],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,9,28]]},"references-count":43,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9,28]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0358480","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,28]]}}}